PCM_FPTPD_Howard_ch5

# Recursive Calculation of First Passage Time Probability Distributions, 
#     Howard, Dynamic Probabilistic Systems, Vol. I, Markov Models, 1971, p.294
#     (5.2.35)
#     R-code by PCM 2015/05/10

fptd <- function(i,j,P,maxT=500)
  # start  state x(0) = i 
  # target state x(t) = j
  # P = n x n transition matrix
  # maxT = maximal time index
{
  n <- nrow(P)
  F <- matrix(0,nrow=n,ncol=n) # fpt(t=0) = initial First Passage Time Probability Distribution F(0)
  F_ij <- c(rep(0,maxT))       # fpt(i,j,t=0:t=maxT) = 0  initial probability distribution
  cum_F_ij <- c(rep(0,maxT))   # fpt(i,j,t=0:t=maxT) = 0  initial cumulative probability distribution
  for (t in 1:maxT)
  {
    F <- if (t == 1) {P} else {P %*% (F - diag(diag(F),nrow=n,ncol=n))}            #        (5.2.35)
    F_ij[t] <- F[i,j]                                 # f[i,j](t) ; t = 1,2,3, ..., maxT    (5.2.35)
    cum_F_ij[t] <- if(t==1) { F_ij[t] } else { cum_F_ij[t-1] +  F_ij[t] }
  }
  barplot(F_ij, names.arg=c(0:(maxT-1)), col="steelblue", 
          main=sprintf("First Pass.Time Prob.Distr., maxT=%i", maxT), xlab="time steps t", 
          ylab=sprintf("pr(s(t)=%i,s(t-1)=/=%i,..,s(1)=/=%i | s(0)=%i)",j,j,j,i), ylim=c(0,1))
  barplot(cum_F_ij, names.arg=c(0:(maxT-1)), col="steelblue", 
          main=sprintf("Cum.First Pass.Time Prob.Distr., maxT=%i", maxT), xlab="time steps t", 
          ylab=sprintf("pr(s(t)=%i,s(t-1)=/=%i,..,s(1)=/=%i | s(0)=%i)",j,j,j,i), ylim=c(0,1))
  return(list(F_ij,cum_F_ij))
}

# Marketing example (Howard, 1971, p.9)
# Results in Table 5.2.1 (Howard, 1971, p.297)
#
P <- matrix(c(0.8, 0.2, 0.3, 0.7),nrow=2,byrow=TRUE) # (1.2.15), p.9
#
fptd(1,2,P,10)

## [[1]]
##  [1] 0.20000000 0.16000000 0.12800000 0.10240000 0.08192000 0.06553600
##  [7] 0.05242880 0.04194304 0.03355443 0.02684355
## 
## [[2]]
##  [1] 0.2000000 0.3600000 0.4880000 0.5904000 0.6723200 0.7378560 0.7902848
##  [8] 0.8322278 0.8657823 0.8926258

#
# #---------------------------------------------------------------------------------------------
# Taxicab example (Howard, 1971, p.24)
P <- matrix(c(0.3,0.2,0.5,0.1,0.8,0.1,0.4,0.4,0.2),nrow=3,byrow=TRUE) # (5.2.70), p.305
fptd(1,3,P,30)

## [[1]]
##  [1] 0.5000000000 0.1700000000 0.0770000000 0.0473000000 0.0350900000
##  [6] 0.0281930000 0.0232925000 0.0194192900 0.0162368690 0.0135883121
## [11] 0.0113750321 0.0095231067 0.0079729103 0.0066751178 0.0055885894
## [16] 0.0046789224 0.0039173250 0.0032796945 0.0027458525 0.0022989049
## [21] 0.0019247079 0.0016114196 0.0013491258 0.0011295261 0.0009456710
## [26] 0.0007917424 0.0006628690 0.0005549726 0.0004646386 0.0003890085
## 
## [[2]]
##  [1] 0.5000000 0.6700000 0.7470000 0.7943000 0.8293900 0.8575830 0.8808755
##  [8] 0.9002948 0.9165317 0.9301200 0.9414950 0.9510181 0.9589910 0.9656661
## [15] 0.9712547 0.9759336 0.9798510 0.9831307 0.9858765 0.9881754 0.9901001
## [22] 0.9917116 0.9930607 0.9941902 0.9951359 0.9959276 0.9965905 0.9971455
## [29] 0.9976101 0.9979991

# MacKay's Example and Rule of Thumb
# P = transition matrix P(State=j at t+1 | State=i at t) = P[i, j]
P <- matrix(c(
  .50,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 01
  .50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 02
  .00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 03
  .00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 04
  .00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 05
  .00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 06
  .00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 07
  .00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 08
  .00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 09
  .00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 10
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,.00,  # row 11
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,.00,  # row 12
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,.00,  # row 13
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,.00,  # row 14
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,.00,  # row 15
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,.00,  # row 16
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,.00,  # row 17
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,.00,  # row 18
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,.00,  # row 19
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.00,.50,  # row 20
  .00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.00,.50,.50)  # row 21
  , nrow=21, ncol=21, byrow=TRUE)
#
fptd(11,21,P,121)

## [[1]]
##   [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
##   [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 9.765625e-04
##  [11] 0.000000e+00 2.441406e-03 0.000000e+00 3.967285e-03 0.000000e+00
##  [16] 5.340576e-03 0.000000e+00 6.484985e-03 0.000000e+00 7.392883e-03
##  [21] 0.000000e+00 8.085966e-03 0.000000e+00 8.595586e-03 0.000000e+00
##  [26] 8.953735e-03 0.000000e+00 9.189360e-03 0.000000e+00 9.327200e-03
##  [31] 4.656613e-10 9.387766e-03 3.608875e-09 9.387766e-03 1.533772e-08
##  [36] 9.340671e-03 4.736648e-08 9.257272e-03 1.189237e-07 9.146185e-03
##  [41] 2.576680e-07 9.014268e-03 4.996671e-07 8.866976e-03 8.885057e-07
##  [46] 8.708638e-03 1.473723e-06 8.542684e-03 2.308833e-06 8.371830e-03
##  [51] 3.449172e-06 8.198220e-03 4.949786e-06 8.023542e-03 6.863511e-06
##  [56] 7.849118e-03 9.239341e-06 7.675975e-03 1.212114e-05 7.504911e-03
##  [61] 1.554667e-05 7.336531e-03 1.954704e-05 7.171294e-03 2.414634e-05
##  [66] 7.009536e-03 2.936160e-05 6.851496e-03 3.520299e-05 6.697337e-03
##  [71] 4.167407e-05 6.547161e-03 4.877230e-05 6.401019e-03 5.648954e-05
##  [76] 6.258924e-03 6.481261e-05 6.120860e-03 7.372393e-05 5.986784e-03
##  [81] 8.320210e-05 5.856636e-03 9.322250e-05 5.730343e-03 1.037578e-04
##  [86] 5.607819e-03 1.147787e-04 5.488972e-03 1.262541e-04 5.373704e-03
##  [91] 1.381518e-04 5.261912e-03 1.504388e-04 5.153494e-03 1.630816e-04
##  [96] 5.048344e-03 1.760468e-04 4.946357e-03 1.893009e-04 4.847430e-03
## [101] 2.028110e-04 4.751460e-03 2.165447e-04 4.658346e-03 2.304705e-04
## [106] 4.567990e-03 2.445576e-04 4.480296e-03 2.587763e-04 4.395170e-03
## [111] 2.730982e-04 4.312523e-03 2.874958e-04 4.232265e-03 3.019428e-04
## [116] 4.154312e-03 3.164145e-04 4.078582e-03 3.308871e-04 4.004997e-03
## [121] 3.453383e-04
## 
## [[2]]
##   [1] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000
##   [6] 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0009765625
##  [11] 0.0009765625 0.0034179688 0.0034179688 0.0073852539 0.0073852539
##  [16] 0.0127258301 0.0127258301 0.0192108154 0.0192108154 0.0266036987
##  [21] 0.0266036987 0.0346896648 0.0346896648 0.0432852507 0.0432852507
##  [26] 0.0522389859 0.0522389859 0.0614283457 0.0614283457 0.0707555460
##  [31] 0.0707555464 0.0801433129 0.0801433165 0.0895310830 0.0895310983
##  [36] 0.0988717690 0.0988718164 0.1081290882 0.1081292071 0.1172753917
##  [41] 0.1172756493 0.1262899178 0.1262904174 0.1351573939 0.1351582824
##  [46] 0.1438669200 0.1438683937 0.1524110772 0.1524133861 0.1607852159
##  [51] 0.1607886651 0.1689868856 0.1689918353 0.1770153778 0.1770222413
##  [56] 0.1848713590 0.1848805983 0.1925565737 0.1925686948 0.2000736056
##  [61] 0.2000891522 0.2074256836 0.2074452306 0.2146165248 0.2146406711
##  [66] 0.2216502067 0.2216795683 0.2285310641 0.2285662670 0.2352636041
##  [71] 0.2353052782 0.2418524389 0.2419012112 0.2483022300 0.2483587195
##  [76] 0.2546176437 0.2546824564 0.2608033161 0.2608770400 0.2668638238
##  [81] 0.2669470259 0.2728036622 0.2728968847 0.2786272278 0.2787309857
##  [86] 0.2843388050 0.2844535837 0.2899425560 0.2900688101 0.2954425140
##  [91] 0.2955806658 0.3008425782 0.3009930170 0.3061465108 0.3063095924
##  [96] 0.3113579359 0.3115339827 0.3164803395 0.3166696404 0.3215170701
## [101] 0.3217198811 0.3264713407 0.3266878854 0.3313462313 0.3315767018
## [106] 0.3361446918 0.3363892494 0.3408695454 0.3411283217 0.3455234921
## [111] 0.3457965903 0.3501091128 0.3503966085 0.3546288732 0.3549308160
## [116] 0.3590851279 0.3594015424 0.3634801247 0.3638110118 0.3678160085
## [121] 0.3681613467

fptd( 1,21,P,441)

## [[1]]
##   [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
##   [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
##  [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
##  [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 9.536743e-07
##  [21] 4.768372e-07 4.768372e-06 2.503395e-06 1.370907e-05 7.510185e-06
##  [26] 2.980232e-05 1.695007e-05 5.448237e-05 3.203563e-05 8.847937e-05
##  [31] 5.359808e-05 1.318683e-04 8.205138e-05 1.841970e-04 1.174256e-04
##  [36] 2.446366e-04 1.594356e-04 3.121226e-04 2.075615e-04 3.854714e-04
##  [41] 2.611258e-04 4.634700e-04 3.193598e-04 5.449393e-04 3.814575e-04
##  [46] 6.287761e-04 4.466160e-04 7.139779e-04 5.140641e-04 7.996553e-04
##  [51] 5.830820e-04 8.850352e-04 6.530124e-04 9.694582e-04 7.232669e-04
##  [56] 1.052372e-03 7.933269e-04 1.133324e-03 8.627430e-04 1.211949e-03
##  [61] 9.311312e-04 1.287959e-03 9.981684e-04 1.361139e-03 1.063587e-03
##  [66] 1.431329e-03 1.127171e-03 1.498422e-03 1.188748e-03 1.562355e-03
##  [71] 1.248186e-03 1.623099e-03 1.305386e-03 1.680656e-03 1.360281e-03
##  [76] 1.735052e-03 1.412828e-03 1.786334e-03 1.463008e-03 1.834565e-03
##  [81] 1.510818e-03 1.879820e-03 1.556274e-03 1.922184e-03 1.599402e-03
##  [86] 1.961748e-03 1.640240e-03 1.998611e-03 1.678834e-03 2.032873e-03
##  [91] 1.715237e-03 2.064637e-03 1.749508e-03 2.094006e-03 1.781710e-03
##  [96] 2.121084e-03 1.811909e-03 2.145972e-03 1.840171e-03 2.168773e-03
## [101] 1.866567e-03 2.189585e-03 1.891166e-03 2.208504e-03 1.914037e-03
## [106] 2.225624e-03 1.935250e-03 2.241036e-03 1.954873e-03 2.254827e-03
## [111] 1.972973e-03 2.267081e-03 1.989617e-03 2.277880e-03 2.004868e-03
## [116] 2.287301e-03 2.018791e-03 2.295419e-03 2.031444e-03 2.302305e-03
## [121] 2.042888e-03 2.308026e-03 2.053179e-03 2.312649e-03 2.062372e-03
## [126] 2.316234e-03 2.070521e-03 2.318841e-03 2.077676e-03 2.320526e-03
## [131] 2.083886e-03 2.321342e-03 2.089199e-03 2.321340e-03 2.093659e-03
## [136] 2.320567e-03 2.097310e-03 2.319070e-03 2.100193e-03 2.316892e-03
## [141] 2.102347e-03 2.314074e-03 2.103810e-03 2.310655e-03 2.104619e-03
## [146] 2.306671e-03 2.104808e-03 2.302159e-03 2.104410e-03 2.297150e-03
## [151] 2.103456e-03 2.291677e-03 2.101976e-03 2.285769e-03 2.099998e-03
## [156] 2.279453e-03 2.097550e-03 2.272758e-03 2.094658e-03 2.265707e-03
## [161] 2.091346e-03 2.258325e-03 2.087638e-03 2.250634e-03 2.083556e-03
## [166] 2.242656e-03 2.079121e-03 2.234410e-03 2.074353e-03 2.225915e-03
## [171] 2.069272e-03 2.217190e-03 2.063896e-03 2.208251e-03 2.058242e-03
## [176] 2.199115e-03 2.052327e-03 2.189797e-03 2.046167e-03 2.180310e-03
## [181] 2.039776e-03 2.170668e-03 2.033168e-03 2.160885e-03 2.026357e-03
## [186] 2.150972e-03 2.019356e-03 2.140940e-03 2.012177e-03 2.130801e-03
## [191] 2.004831e-03 2.120563e-03 1.997330e-03 2.110238e-03 1.989684e-03
## [196] 2.099833e-03 1.981902e-03 2.089358e-03 1.973994e-03 2.078820e-03
## [201] 1.965969e-03 2.068227e-03 1.957835e-03 2.057585e-03 1.949601e-03
## [206] 2.046902e-03 1.941274e-03 2.036184e-03 1.932861e-03 2.025437e-03
## [211] 1.924369e-03 2.014666e-03 1.915804e-03 2.003877e-03 1.907173e-03
## [216] 1.993074e-03 1.898482e-03 1.982263e-03 1.889736e-03 1.971447e-03
## [221] 1.880940e-03 1.960630e-03 1.872099e-03 1.949817e-03 1.863218e-03
## [226] 1.939011e-03 1.854302e-03 1.928216e-03 1.845355e-03 1.917434e-03
## [231] 1.836381e-03 1.906669e-03 1.827383e-03 1.895923e-03 1.818365e-03
## [236] 1.885199e-03 1.809331e-03 1.874500e-03 1.800284e-03 1.863827e-03
## [241] 1.791227e-03 1.853184e-03 1.782163e-03 1.842571e-03 1.773095e-03
## [246] 1.831991e-03 1.764024e-03 1.821445e-03 1.754955e-03 1.810935e-03
## [251] 1.745888e-03 1.800463e-03 1.736827e-03 1.790030e-03 1.727773e-03
## [256] 1.779637e-03 1.718729e-03 1.769286e-03 1.709696e-03 1.758977e-03
## [261] 1.700675e-03 1.748711e-03 1.691670e-03 1.738491e-03 1.682680e-03
## [266] 1.728315e-03 1.673709e-03 1.718186e-03 1.664756e-03 1.708104e-03
## [271] 1.655824e-03 1.698069e-03 1.646914e-03 1.688082e-03 1.638026e-03
## [276] 1.678144e-03 1.629163e-03 1.668256e-03 1.620324e-03 1.658417e-03
## [281] 1.611512e-03 1.648628e-03 1.602727e-03 1.638890e-03 1.593969e-03
## [286] 1.629202e-03 1.585240e-03 1.619565e-03 1.576540e-03 1.609980e-03
## [291] 1.567871e-03 1.600446e-03 1.559232e-03 1.590963e-03 1.550625e-03
## [296] 1.581532e-03 1.542050e-03 1.572154e-03 1.533508e-03 1.562827e-03
## [301] 1.524998e-03 1.553552e-03 1.516523e-03 1.544329e-03 1.508081e-03
## [306] 1.535158e-03 1.499674e-03 1.526040e-03 1.491301e-03 1.516973e-03
## [311] 1.482964e-03 1.507958e-03 1.474663e-03 1.498995e-03 1.466397e-03
## [316] 1.490084e-03 1.458167e-03 1.481225e-03 1.449974e-03 1.472417e-03
## [321] 1.441817e-03 1.463661e-03 1.433698e-03 1.454956e-03 1.425615e-03
## [326] 1.446302e-03 1.417569e-03 1.437700e-03 1.409561e-03 1.429148e-03
## [331] 1.401591e-03 1.420647e-03 1.393658e-03 1.412196e-03 1.385763e-03
## [336] 1.403796e-03 1.377905e-03 1.395446e-03 1.370086e-03 1.387145e-03
## [341] 1.362305e-03 1.378895e-03 1.354561e-03 1.370694e-03 1.346856e-03
## [346] 1.362542e-03 1.339189e-03 1.354439e-03 1.331560e-03 1.346385e-03
## [351] 1.323969e-03 1.338379e-03 1.316417e-03 1.330422e-03 1.308902e-03
## [356] 1.322513e-03 1.301426e-03 1.314651e-03 1.293988e-03 1.306837e-03
## [361] 1.286588e-03 1.299071e-03 1.279226e-03 1.291351e-03 1.271902e-03
## [366] 1.283678e-03 1.264617e-03 1.276052e-03 1.257368e-03 1.268472e-03
## [371] 1.250158e-03 1.260938e-03 1.242986e-03 1.253449e-03 1.235851e-03
## [376] 1.246006e-03 1.228754e-03 1.238609e-03 1.221695e-03 1.231256e-03
## [381] 1.214673e-03 1.223948e-03 1.207688e-03 1.216684e-03 1.200740e-03
## [386] 1.209464e-03 1.193830e-03 1.202289e-03 1.186957e-03 1.195157e-03
## [391] 1.180120e-03 1.188068e-03 1.173321e-03 1.181022e-03 1.166558e-03
## [396] 1.174019e-03 1.159832e-03 1.167059e-03 1.153142e-03 1.160141e-03
## [401] 1.146488e-03 1.153265e-03 1.139871e-03 1.146431e-03 1.133290e-03
## [406] 1.139638e-03 1.126745e-03 1.132886e-03 1.120235e-03 1.126176e-03
## [411] 1.113762e-03 1.119506e-03 1.107324e-03 1.112876e-03 1.100921e-03
## [416] 1.106287e-03 1.094553e-03 1.099738e-03 1.088221e-03 1.093228e-03
## [421] 1.081924e-03 1.086758e-03 1.075661e-03 1.080327e-03 1.069433e-03
## [426] 1.073935e-03 1.063240e-03 1.067582e-03 1.057081e-03 1.061267e-03
## [431] 1.050956e-03 1.054991e-03 1.044866e-03 1.048752e-03 1.038809e-03
## [436] 1.042551e-03 1.032787e-03 1.036388e-03 1.026798e-03 1.030262e-03
## [441] 1.020842e-03
## 
## [[2]]
##   [1] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
##   [6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
##  [11] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
##  [16] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 9.536743e-07
##  [21] 1.430511e-06 6.198883e-06 8.702278e-06 2.241135e-05 2.992153e-05
##  [26] 5.972385e-05 7.667392e-05 1.311563e-04 1.631919e-04 2.516713e-04
##  [31] 3.052694e-04 4.371377e-04 5.191891e-04 7.033860e-04 8.208116e-04
##  [36] 1.065448e-03 1.224884e-03 1.537006e-03 1.744568e-03 2.130039e-03
##  [41] 2.391165e-03 2.854635e-03 3.173995e-03 3.718934e-03 4.100392e-03
##  [46] 4.729168e-03 5.175784e-03 5.889762e-03 6.403826e-03 7.203481e-03
##  [51] 7.786563e-03 8.671598e-03 9.324611e-03 1.029407e-02 1.101734e-02
##  [56] 1.206971e-02 1.286304e-02 1.399636e-02 1.485910e-02 1.607105e-02
##  [61] 1.700218e-02 1.829014e-02 1.928831e-02 2.064945e-02 2.171304e-02
##  [66] 2.314436e-02 2.427154e-02 2.576996e-02 2.695871e-02 2.852106e-02
##  [71] 2.976925e-02 3.139235e-02 3.269773e-02 3.437839e-02 3.573867e-02
##  [76] 3.747372e-02 3.888655e-02 4.067288e-02 4.213589e-02 4.397046e-02
##  [81] 4.548127e-02 4.736109e-02 4.891737e-02 5.083955e-02 5.243895e-02
##  [86] 5.440070e-02 5.604094e-02 5.803955e-02 5.971839e-02 6.175126e-02
##  [91] 6.346650e-02 6.553113e-02 6.728064e-02 6.937465e-02 7.115636e-02
##  [96] 7.327744e-02 7.508935e-02 7.723532e-02 7.907549e-02 8.124427e-02
## [101] 8.311083e-02 8.530042e-02 8.719158e-02 8.940009e-02 9.131413e-02
## [106] 9.353975e-02 9.547500e-02 9.771604e-02 9.967091e-02 1.019257e-01
## [111] 1.038987e-01 1.061658e-01 1.081554e-01 1.104333e-01 1.124382e-01
## [116] 1.147255e-01 1.167442e-01 1.190397e-01 1.210711e-01 1.233734e-01
## [121] 1.254163e-01 1.277243e-01 1.297775e-01 1.320902e-01 1.341525e-01
## [126] 1.364688e-01 1.385393e-01 1.408581e-01 1.429358e-01 1.452563e-01
## [131] 1.473402e-01 1.496616e-01 1.517508e-01 1.540721e-01 1.561658e-01
## [136] 1.584863e-01 1.605836e-01 1.629027e-01 1.650029e-01 1.673198e-01
## [141] 1.694221e-01 1.717362e-01 1.738400e-01 1.761507e-01 1.782553e-01
## [146] 1.805620e-01 1.826668e-01 1.849689e-01 1.870733e-01 1.893705e-01
## [151] 1.914739e-01 1.937656e-01 1.958676e-01 1.981534e-01 2.002534e-01
## [156] 2.025328e-01 2.046304e-01 2.069031e-01 2.089978e-01 2.112635e-01
## [161] 2.133548e-01 2.156132e-01 2.177008e-01 2.199514e-01 2.220350e-01
## [166] 2.242776e-01 2.263568e-01 2.285912e-01 2.306655e-01 2.328914e-01
## [171] 2.349607e-01 2.371779e-01 2.392418e-01 2.414501e-01 2.435083e-01
## [176] 2.457074e-01 2.477597e-01 2.499495e-01 2.519957e-01 2.541760e-01
## [181] 2.562158e-01 2.583865e-01 2.604196e-01 2.625805e-01 2.646069e-01
## [186] 2.667578e-01 2.687772e-01 2.709181e-01 2.729303e-01 2.750611e-01
## [191] 2.770659e-01 2.791865e-01 2.811838e-01 2.832941e-01 2.852838e-01
## [196] 2.873836e-01 2.893655e-01 2.914549e-01 2.934288e-01 2.955077e-01
## [201] 2.974736e-01 2.995419e-01 3.014997e-01 3.035573e-01 3.055069e-01
## [206] 3.075538e-01 3.094951e-01 3.115312e-01 3.134641e-01 3.154895e-01
## [211] 3.174139e-01 3.194286e-01 3.213444e-01 3.233483e-01 3.252554e-01
## [216] 3.272485e-01 3.291470e-01 3.311293e-01 3.330190e-01 3.349904e-01
## [221] 3.368714e-01 3.388320e-01 3.407041e-01 3.426539e-01 3.445171e-01
## [226] 3.464562e-01 3.483105e-01 3.502387e-01 3.520840e-01 3.540015e-01
## [231] 3.558378e-01 3.577445e-01 3.595719e-01 3.614678e-01 3.632862e-01
## [236] 3.651714e-01 3.669807e-01 3.688552e-01 3.706555e-01 3.725193e-01
## [241] 3.743105e-01 3.761637e-01 3.779459e-01 3.797885e-01 3.815616e-01
## [246] 3.833936e-01 3.851576e-01 3.869790e-01 3.887340e-01 3.905449e-01
## [251] 3.922908e-01 3.940913e-01 3.958281e-01 3.976181e-01 3.993459e-01
## [256] 4.011255e-01 4.028443e-01 4.046135e-01 4.063232e-01 4.080822e-01
## [261] 4.097829e-01 4.115316e-01 4.132233e-01 4.149618e-01 4.166444e-01
## [266] 4.183728e-01 4.200465e-01 4.217647e-01 4.234294e-01 4.251375e-01
## [271] 4.267933e-01 4.284914e-01 4.301383e-01 4.318264e-01 4.334644e-01
## [276] 4.351426e-01 4.367717e-01 4.384400e-01 4.400603e-01 4.417187e-01
## [281] 4.433302e-01 4.449789e-01 4.465816e-01 4.482205e-01 4.498145e-01
## [286] 4.514437e-01 4.530289e-01 4.546485e-01 4.562250e-01 4.578350e-01
## [291] 4.594029e-01 4.610033e-01 4.625625e-01 4.641535e-01 4.657041e-01
## [296] 4.672857e-01 4.688277e-01 4.703999e-01 4.719334e-01 4.734962e-01
## [301] 4.750212e-01 4.765747e-01 4.780913e-01 4.796356e-01 4.811437e-01
## [306] 4.826788e-01 4.841785e-01 4.857045e-01 4.871958e-01 4.887128e-01
## [311] 4.901958e-01 4.917037e-01 4.931784e-01 4.946774e-01 4.961438e-01
## [316] 4.976339e-01 4.990920e-01 5.005733e-01 5.020232e-01 5.034957e-01
## [321] 5.049375e-01 5.064011e-01 5.078348e-01 5.092898e-01 5.107154e-01
## [326] 5.121617e-01 5.135793e-01 5.150170e-01 5.164265e-01 5.178557e-01
## [331] 5.192573e-01 5.206779e-01 5.220716e-01 5.234838e-01 5.248695e-01
## [336] 5.262733e-01 5.276512e-01 5.290467e-01 5.304168e-01 5.318039e-01
## [341] 5.331662e-01 5.345451e-01 5.358997e-01 5.372704e-01 5.386172e-01
## [346] 5.399798e-01 5.413190e-01 5.426734e-01 5.440050e-01 5.453514e-01
## [351] 5.466753e-01 5.480137e-01 5.493301e-01 5.506605e-01 5.519694e-01
## [356] 5.532920e-01 5.545934e-01 5.559080e-01 5.572020e-01 5.585089e-01
## [361] 5.597954e-01 5.610945e-01 5.623737e-01 5.636651e-01 5.649370e-01
## [366] 5.662207e-01 5.674853e-01 5.687613e-01 5.700187e-01 5.712872e-01
## [371] 5.725373e-01 5.737983e-01 5.750413e-01 5.762947e-01 5.775306e-01
## [376] 5.787766e-01 5.800053e-01 5.812439e-01 5.824656e-01 5.836969e-01
## [381] 5.849116e-01 5.861355e-01 5.873432e-01 5.885599e-01 5.897606e-01
## [386] 5.909701e-01 5.921639e-01 5.933662e-01 5.945532e-01 5.957483e-01
## [391] 5.969284e-01 5.981165e-01 5.992898e-01 6.004708e-01 6.016374e-01
## [396] 6.028114e-01 6.039713e-01 6.051383e-01 6.062915e-01 6.074516e-01
## [401] 6.085981e-01 6.097513e-01 6.108912e-01 6.120376e-01 6.131709e-01
## [406] 6.143106e-01 6.154373e-01 6.165702e-01 6.176904e-01 6.188166e-01
## [411] 6.199304e-01 6.210499e-01 6.221572e-01 6.232701e-01 6.243710e-01
## [416] 6.254773e-01 6.265718e-01 6.276716e-01 6.287598e-01 6.298530e-01
## [421] 6.309350e-01 6.320217e-01 6.330974e-01 6.341777e-01 6.352471e-01
## [426] 6.363211e-01 6.373843e-01 6.384519e-01 6.395090e-01 6.405702e-01
## [431] 6.416212e-01 6.426762e-01 6.437211e-01 6.447698e-01 6.458086e-01
## [436] 6.468512e-01 6.478840e-01 6.489203e-01 6.499471e-01 6.509774e-01
## [441] 6.519982e-01

PCM_FPTPD_Howard_ch5_2.R

PCM

Sun Feb 21 21:38:39 2016